ProjectAn alternative development of analytic number theory and applications

Researcher (PI)ANDREW Granville

Host Institution (HI)UNIVERSITY COLLEGE LONDON

Call DetailsAdvanced Grant (AdG), PE1, ERC-2014-ADG

SummaryThe traditional (Riemann) approach to analytic number theory uses the zeros of zeta functions. This requires the associated multiplicative function, say f(n), to have special enough properties that the associated Dirichlet series may be analytically continued. In this proposal we continue to develop an approach which requires less of the multiplicative function, linking the original question with the mean value of f. Such techniques have been around for a long time but have generally been regarded as “ad hoc”. In this project we aim to show that one can develop a coherent approach to the whole subject, not only reproving all of the old results, but also many new ones that appear inaccessible to traditional methods.
Our first goal is to complete a monograph yielding a reworking of all the classical theory using these new methods and then to push forward in new directions. The most important is to extend these techniques to GL(n) L-functions, which we hope will now be feasible having found the correct framework in which to proceed. Since we rarely know how to analytically continue such L-functions this could be of great benefit to the subject.
We are developing the large sieve so that it can be used for individual moduli, and will determine a strong form of that. Also a new method to give asymptotics for mean values, when they are not too small.
We wish to incorporate techniques of analytic number theory into our theory, for example recent advances on mean values of Dirichlet polynomials. Also the recent breakthroughs on the sieve suggest strong links that need further exploration.
Additive combinatorics yields important results in many areas. There are strong analogies between its results, and those for multiplicative functions, especially in large value spectrum theory, and its applications. We hope to develop these further.
Much of this is joint work with K Soundararajan of Stanford University.

The traditional (Riemann) approach to analytic number theory uses the zeros of zeta functions. This requires the associated multiplicative function, say f(n), to have special enough properties that the associated Dirichlet series may be analytically continued. In this proposal we continue to develop an approach which requires less of the multiplicative function, linking the original question with the mean value of f. Such techniques have been around for a long time but have generally been regarded as “ad hoc”. In this project we aim to show that one can develop a coherent approach to the whole subject, not only reproving all of the old results, but also many new ones that appear inaccessible to traditional methods.
Our first goal is to complete a monograph yielding a reworking of all the classical theory using these new methods and then to push forward in new directions. The most important is to extend these techniques to GL(n) L-functions, which we hope will now be feasible having found the correct framework in which to proceed. Since we rarely know how to analytically continue such L-functions this could be of great benefit to the subject.
We are developing the large sieve so that it can be used for individual moduli, and will determine a strong form of that. Also a new method to give asymptotics for mean values, when they are not too small.
We wish to incorporate techniques of analytic number theory into our theory, for example recent advances on mean values of Dirichlet polynomials. Also the recent breakthroughs on the sieve suggest strong links that need further exploration.
Additive combinatorics yields important results in many areas. There are strong analogies between its results, and those for multiplicative functions, especially in large value spectrum theory, and its applications. We hope to develop these further.
Much of this is joint work with K Soundararajan of Stanford University.

Max ERC Funding

2 011 742 €

Duration

Start date: 2015-08-01, End date: 2020-07-31

Project acronym2-3-AUT

ProjectSurfaces, 3-manifolds and automorphism groups

Researcher (PI)Nathalie Wahl

Host Institution (HI)KOBENHAVNS UNIVERSITET

Call DetailsStarting Grant (StG), PE1, ERC-2009-StG

SummaryThe scientific goal of the proposal is to answer central questions related to diffeomorphism groups of manifolds of dimension 2 and 3, and to their deformation invariant analogs, the mapping class groups. While the classification of surfaces has been known for more than a century, their automorphism groups have yet to be fully understood. Even less is known about diffeomorphisms of 3-manifolds despite much interest, and the objects here have only been classified recently, by the breakthrough work of Perelman on the Poincar\&apos;e and geometrization conjectures. In dimension 2, I will focus on the relationship between mapping class groups and topological conformal field theories, with applications to Hochschild homology. In dimension 3, I propose to compute the stable homology of classifying spaces of diffeomorphism groups and mapping class groups, as well as study the homotopy type of the space of diffeomorphisms. I propose moreover to establish homological stability theorems in the wider context of automorphism groups and more general families of groups. The project combines breakthrough methods from homotopy theory with methods from differential and geometric topology. The research team will consist of 3 PhD students, and 4 postdocs, which I will lead.

The scientific goal of the proposal is to answer central questions related to diffeomorphism groups of manifolds of dimension 2 and 3, and to their deformation invariant analogs, the mapping class groups. While the classification of surfaces has been known for more than a century, their automorphism groups have yet to be fully understood. Even less is known about diffeomorphisms of 3-manifolds despite much interest, and the objects here have only been classified recently, by the breakthrough work of Perelman on the Poincar\&apos;e and geometrization conjectures. In dimension 2, I will focus on the relationship between mapping class groups and topological conformal field theories, with applications to Hochschild homology. In dimension 3, I propose to compute the stable homology of classifying spaces of diffeomorphism groups and mapping class groups, as well as study the homotopy type of the space of diffeomorphisms. I propose moreover to establish homological stability theorems in the wider context of automorphism groups and more general families of groups. The project combines breakthrough methods from homotopy theory with methods from differential and geometric topology. The research team will consist of 3 PhD students, and 4 postdocs, which I will lead.

SummaryThe applicant will collaborate with Irish, European and U.S.-based colleagues to develop a sustainable biorefinery and bioenergy industry in Ireland and Europe. The focus of this ERC Starting Grant will be the application of classical microbiological, physiological and real-time polymerase chain reaction (PCR)-based assays, to qualitatively and quantitatively characterize microbial communities underpinning novel and innovative, low-temperature, anaerobic waste (and other biomass) conversion technologies, including municipal wastewater treatment and, demonstration- and full-scale biorefinery applications.
Anaerobic digestion (AD) is a naturally-occurring process, which is widely applied for the conversion of waste to methane-containing biogas. Low-temperature (<20 degrees C) AD has been applied by the applicant as a cost-effective alternative to mesophilic (c. 35C) AD for the treatment of several waste categories. However, the microbiology of low-temperature AD is poorly understood. The applicant will work with microbial consortia isolated from anaerobic bioreactors, which have been operated for long-term experiments (>3.5 years), and include organic acid-oxidizing, hydrogen-producing syntrophic microbes and hydrogen-consuming methanogens. A major focus of the project will be the ecophysiology of psychrotolerant and psychrophilic methanogens already identified and cultivated by the applicant. The project will also investigate the role(s) of poorly-understood Crenarchaeota populations and homoacetogenic bacteria, in complex consortia. The host organization is a leading player in the microbiology of waste-to-energy applications. The applicant will train a team of scientists in all aspects of the microbiology and bioengineering of biomass conversion systems.

The applicant will collaborate with Irish, European and U.S.-based colleagues to develop a sustainable biorefinery and bioenergy industry in Ireland and Europe. The focus of this ERC Starting Grant will be the application of classical microbiological, physiological and real-time polymerase chain reaction (PCR)-based assays, to qualitatively and quantitatively characterize microbial communities underpinning novel and innovative, low-temperature, anaerobic waste (and other biomass) conversion technologies, including municipal wastewater treatment and, demonstration- and full-scale biorefinery applications.
Anaerobic digestion (AD) is a naturally-occurring process, which is widely applied for the conversion of waste to methane-containing biogas. Low-temperature (<20 degrees C) AD has been applied by the applicant as a cost-effective alternative to mesophilic (c. 35C) AD for the treatment of several waste categories. However, the microbiology of low-temperature AD is poorly understood. The applicant will work with microbial consortia isolated from anaerobic bioreactors, which have been operated for long-term experiments (>3.5 years), and include organic acid-oxidizing, hydrogen-producing syntrophic microbes and hydrogen-consuming methanogens. A major focus of the project will be the ecophysiology of psychrotolerant and psychrophilic methanogens already identified and cultivated by the applicant. The project will also investigate the role(s) of poorly-understood Crenarchaeota populations and homoacetogenic bacteria, in complex consortia. The host organization is a leading player in the microbiology of waste-to-energy applications. The applicant will train a team of scientists in all aspects of the microbiology and bioengineering of biomass conversion systems.

SummaryChronic lung diseases are increasing in prevalence with over 65 million patients worldwide. Lung transplantation remains the only potential option at end-stage disease. Around 4000 patients receive lung transplants annually with more awaiting transplantation, including 1000 patients in Europe. New options to increase available tissue for lung transplantation are desperately needed.
An exciting new research area focuses on generating lung tissue ex vivo using bioengineering approaches. Scaffolds can be generated from synthetic or biologically-derived (acellular) materials, seeded with cells and grown in a bioreactor prior to transplantation. Ideally, scaffolds would be seeded with cells derived from the transplant recipient, thus obviating the need for long-term immunosuppression. However, functional regeneration has yet to be achieved. New advances in 3D printing and 3D bioprinting (when cells are printed) indicate that this once thought of science-fiction concept might finally be mature enough for complex tissues, including lung. 3D bioprinting addresses a number of concerns identified in previous approaches, such as a) patient heterogeneity in acellular human scaffolds, b) anatomical differences in xenogeneic sources, c) lack of biological cues on synthetic materials and d) difficulty in manufacturing the complex lung architecture. 3D bioprinting could be a reproducible, scalable, and controllable approach for generating functional lung tissue.
The aim of this proposal is to use custom 3D bioprinters to generate constructs mimicking lung tissue using an innovative approach combining primary cells, the engineering reproducibility of synthetic materials, and the biologically conductive properties of acellular lung (hybrid). We will 3D bioprint hybrid murine and human lung tissue models and test gas exchange, angiogenesis and in vivo immune responses. This proposal will be a critical first step in demonstrating feasibility of 3D bioprinting lung tissue.

Chronic lung diseases are increasing in prevalence with over 65 million patients worldwide. Lung transplantation remains the only potential option at end-stage disease. Around 4000 patients receive lung transplants annually with more awaiting transplantation, including 1000 patients in Europe. New options to increase available tissue for lung transplantation are desperately needed.
An exciting new research area focuses on generating lung tissue ex vivo using bioengineering approaches. Scaffolds can be generated from synthetic or biologically-derived (acellular) materials, seeded with cells and grown in a bioreactor prior to transplantation. Ideally, scaffolds would be seeded with cells derived from the transplant recipient, thus obviating the need for long-term immunosuppression. However, functional regeneration has yet to be achieved. New advances in 3D printing and 3D bioprinting (when cells are printed) indicate that this once thought of science-fiction concept might finally be mature enough for complex tissues, including lung. 3D bioprinting addresses a number of concerns identified in previous approaches, such as a) patient heterogeneity in acellular human scaffolds, b) anatomical differences in xenogeneic sources, c) lack of biological cues on synthetic materials and d) difficulty in manufacturing the complex lung architecture. 3D bioprinting could be a reproducible, scalable, and controllable approach for generating functional lung tissue.
The aim of this proposal is to use custom 3D bioprinters to generate constructs mimicking lung tissue using an innovative approach combining primary cells, the engineering reproducibility of synthetic materials, and the biologically conductive properties of acellular lung (hybrid). We will 3D bioprint hybrid murine and human lung tissue models and test gas exchange, angiogenesis and in vivo immune responses. This proposal will be a critical first step in demonstrating feasibility of 3D bioprinting lung tissue.

Max ERC Funding

1 499 975 €

Duration

Start date: 2019-01-01, End date: 2023-12-31

Project acronym3DPROTEINPUZZLES

ProjectShape-directed protein assembly design

Researcher (PI)Lars Ingemar ANDRÉ

Host Institution (HI)LUNDS UNIVERSITET

Call DetailsConsolidator Grant (CoG), LS9, ERC-2017-COG

SummaryLarge protein complexes carry out some of the most complex functions in biology. Such structures are often assembled spontaneously from individual components through the process of self-assembly. If self-assembled protein complexes could be engineered from first principle it would enable a wide range of applications in biomedicine, nanotechnology and materials science. Recently, approaches to rationally design proteins to self-assembly into predefined structures have emerged. The highlight of this work is the design of protein cages that may be engineered into protein containers. However, current approaches for self-assembly design does not result in the assemblies with the required structural complexity to encode many of the sophisticated functions found in nature. To move forward, we have to learn how to engineer protein subunits with more than one designed interface that can assemble into tightly interacting complexes. In this proposal we propose a new protein design paradigm, shape directed protein design, in order to address shortcomings of the current methodology. The proposed method combines geometric shape matching and computational protein design. Using this approach we will de novo design assemblies with a wide variety of structural states, including protein complexes with cyclic and dihedral symmetry as well as icosahedral protein capsids built from novel protein building blocks. To enable these two design challenges we also develop a high-throughput assay to measure assembly stability in vivo that builds on a three-color fluorescent assay. This method will not only facilitate the screening of orders of magnitude more design constructs, but also enable the application of directed evolution to experimentally improve stable and assembly properties of designed containers as well as other designed assemblies.

Large protein complexes carry out some of the most complex functions in biology. Such structures are often assembled spontaneously from individual components through the process of self-assembly. If self-assembled protein complexes could be engineered from first principle it would enable a wide range of applications in biomedicine, nanotechnology and materials science. Recently, approaches to rationally design proteins to self-assembly into predefined structures have emerged. The highlight of this work is the design of protein cages that may be engineered into protein containers. However, current approaches for self-assembly design does not result in the assemblies with the required structural complexity to encode many of the sophisticated functions found in nature. To move forward, we have to learn how to engineer protein subunits with more than one designed interface that can assemble into tightly interacting complexes. In this proposal we propose a new protein design paradigm, shape directed protein design, in order to address shortcomings of the current methodology. The proposed method combines geometric shape matching and computational protein design. Using this approach we will de novo design assemblies with a wide variety of structural states, including protein complexes with cyclic and dihedral symmetry as well as icosahedral protein capsids built from novel protein building blocks. To enable these two design challenges we also develop a high-throughput assay to measure assembly stability in vivo that builds on a three-color fluorescent assay. This method will not only facilitate the screening of orders of magnitude more design constructs, but also enable the application of directed evolution to experimentally improve stable and assembly properties of designed containers as well as other designed assemblies.

Max ERC Funding

2 325 292 €

Duration

Start date: 2018-06-01, End date: 2023-05-31

Project acronym3DWATERWAVES

ProjectMathematical aspects of three-dimensional water waves with vorticity

Researcher (PI)Erik Torsten Wahlén

Host Institution (HI)LUNDS UNIVERSITET

Call DetailsStarting Grant (StG), PE1, ERC-2015-STG

SummaryThe goal of this project is to develop a mathematical theory for steady three-dimensional water waves with vorticity. The mathematical model consists of the incompressible Euler equations with a free surface, and vorticity is important for modelling the interaction of surface waves with non-uniform currents. In the two-dimensional case, there has been a lot of progress on water waves with vorticity in the last decade. This progress has mainly been based on the stream function formulation, in which the problem is reformulated as a nonlinear elliptic free boundary problem. An analogue of this formulation is not available in three dimensions, and the theory has therefore so far been restricted to irrotational flow. In this project we seek to go beyond this restriction using two different approaches. In the first approach we will adapt methods which have been used to construct three-dimensional ideal flows with vorticity in domains with a fixed boundary to the free boundary context (for example Beltrami flows). In the second approach we will develop methods which are new even in the case of a fixed boundary, by performing a detailed study of the structure of the equations close to a given shear flow using ideas from infinite-dimensional bifurcation theory. This involves handling infinitely many resonances.

The goal of this project is to develop a mathematical theory for steady three-dimensional water waves with vorticity. The mathematical model consists of the incompressible Euler equations with a free surface, and vorticity is important for modelling the interaction of surface waves with non-uniform currents. In the two-dimensional case, there has been a lot of progress on water waves with vorticity in the last decade. This progress has mainly been based on the stream function formulation, in which the problem is reformulated as a nonlinear elliptic free boundary problem. An analogue of this formulation is not available in three dimensions, and the theory has therefore so far been restricted to irrotational flow. In this project we seek to go beyond this restriction using two different approaches. In the first approach we will adapt methods which have been used to construct three-dimensional ideal flows with vorticity in domains with a fixed boundary to the free boundary context (for example Beltrami flows). In the second approach we will develop methods which are new even in the case of a fixed boundary, by performing a detailed study of the structure of the equations close to a given shear flow using ideas from infinite-dimensional bifurcation theory. This involves handling infinitely many resonances.

SummaryMucins are a family of high-molecular-weight glycoproteins and a major macromolecular constituent in slimy mucus gels that are covering the surface of internal biological tissues. A primary role of mucus gels in biological systems is known to be the protection and lubrication of underlying epithelial cell surfaces. This is intuitively well appreciated by both science community and the public, and yet detailed lubrication properties of mucins and mucus gels have remained largely unexplored to date. Detailed and systematic understanding of the lubrication mechanism of mucus gels is significant from many angles; firstly, lubricity of mucus gels is closely related with fundamental functions of various human organs, such as eye blinking, mastication in oral cavity, swallowing through esophagus, digestion in stomach, breathing through air way and respiratory organs, and thus often indicates the health state of those organs. Furthermore, for the application of various tissue-contacting devices or personal care products, e.g. catheters, endoscopes, and contact lenses, mucus gel layer is the first counter surface that comes into the mechanical and tribological contacts with them. Finally, remarkable lubricating performance by mucins and mucus gels in biological systems may provide many useful and possibly innovative hints in utilizing water as base lubricant for man-made engineering systems. This project thus proposes to carry out a 5 year research program focusing on exploring the lubricity of mucins and mucus gels by combining a broad range of experimental approaches in biology and tribology.

Mucins are a family of high-molecular-weight glycoproteins and a major macromolecular constituent in slimy mucus gels that are covering the surface of internal biological tissues. A primary role of mucus gels in biological systems is known to be the protection and lubrication of underlying epithelial cell surfaces. This is intuitively well appreciated by both science community and the public, and yet detailed lubrication properties of mucins and mucus gels have remained largely unexplored to date. Detailed and systematic understanding of the lubrication mechanism of mucus gels is significant from many angles; firstly, lubricity of mucus gels is closely related with fundamental functions of various human organs, such as eye blinking, mastication in oral cavity, swallowing through esophagus, digestion in stomach, breathing through air way and respiratory organs, and thus often indicates the health state of those organs. Furthermore, for the application of various tissue-contacting devices or personal care products, e.g. catheters, endoscopes, and contact lenses, mucus gel layer is the first counter surface that comes into the mechanical and tribological contacts with them. Finally, remarkable lubricating performance by mucins and mucus gels in biological systems may provide many useful and possibly innovative hints in utilizing water as base lubricant for man-made engineering systems. This project thus proposes to carry out a 5 year research program focusing on exploring the lubricity of mucins and mucus gels by combining a broad range of experimental approaches in biology and tribology.

Max ERC Funding

1 432 920 €

Duration

Start date: 2011-04-01, End date: 2016-03-31

Project acronymAAMOT

ProjectArithmetic of automorphic motives

Researcher (PI)Michael Harris

Host Institution (HI)INSTITUT DES HAUTES ETUDES SCIENTIFIQUES

Call DetailsAdvanced Grant (AdG), PE1, ERC-2011-ADG_20110209

SummaryThe primary purpose of this project is to build on recent spectacular progress in the Langlands program to study the arithmetic properties of automorphic motives constructed in the cohomology of Shimura varieties. Because automorphic methods are available to study the L-functions of these motives, which include elliptic curves and certain families of Calabi-Yau varieties over totally real fields (possibly after base change), they represent the most accessible class of varieties for which one can hope to verify fundamental conjectures on special values of L-functions, including Deligne's conjecture and the Main Conjecture of Iwasawa theory. Immediate goals include the proof of irreducibility of automorphic Galois representations; the establishment of period relations for automorphic and potentially automorphic realizations of motives in the cohomology of distinct Shimura varieties; the construction of p-adic L-functions for these and related motives, notably adjoint and tensor product L-functions in p-adic families; and the geometrization of the p-adic and mod p Langlands program. All four goals, as well as the others mentioned in the body of the proposal, are interconnected; the final goal provides a bridge to related work in geometric representation theory, algebraic geometry, and mathematical physics.

The primary purpose of this project is to build on recent spectacular progress in the Langlands program to study the arithmetic properties of automorphic motives constructed in the cohomology of Shimura varieties. Because automorphic methods are available to study the L-functions of these motives, which include elliptic curves and certain families of Calabi-Yau varieties over totally real fields (possibly after base change), they represent the most accessible class of varieties for which one can hope to verify fundamental conjectures on special values of L-functions, including Deligne's conjecture and the Main Conjecture of Iwasawa theory. Immediate goals include the proof of irreducibility of automorphic Galois representations; the establishment of period relations for automorphic and potentially automorphic realizations of motives in the cohomology of distinct Shimura varieties; the construction of p-adic L-functions for these and related motives, notably adjoint and tensor product L-functions in p-adic families; and the geometrization of the p-adic and mod p Langlands program. All four goals, as well as the others mentioned in the body of the proposal, are interconnected; the final goal provides a bridge to related work in geometric representation theory, algebraic geometry, and mathematical physics.

Max ERC Funding

1 491 348 €

Duration

Start date: 2012-06-01, End date: 2018-05-31

Project acronymAAS

ProjectApproximate algebraic structure and applications

Researcher (PI)Ben Green

Host Institution (HI)THE CHANCELLOR, MASTERS AND SCHOLARS OF THE UNIVERSITY OF OXFORD

Call DetailsStarting Grant (StG), PE1, ERC-2011-StG_20101014

SummaryThis project studies several mathematical topics with a related theme, all of them part of the relatively new discipline known as additive combinatorics.
We look at approximate, or rough, variants of familiar mathematical notions such as group, polynomial or homomorphism. In each case we seek to describe the structure of these approximate objects, and then to give applications of the resulting theorems. This endeavour has already lead to groundbreaking results in the theory of prime numbers, group theory and combinatorial number theory.

This project studies several mathematical topics with a related theme, all of them part of the relatively new discipline known as additive combinatorics.
We look at approximate, or rough, variants of familiar mathematical notions such as group, polynomial or homomorphism. In each case we seek to describe the structure of these approximate objects, and then to give applications of the resulting theorems. This endeavour has already lead to groundbreaking results in the theory of prime numbers, group theory and combinatorial number theory.

Max ERC Funding

1 000 000 €

Duration

Start date: 2011-10-01, End date: 2016-09-30

Project acronymABEL

Project"Alpha-helical Barrels: Exploring, Understanding and Exploiting a New Class of Protein Structure"

Researcher (PI)Derek Neil Woolfson

Host Institution (HI)UNIVERSITY OF BRISTOL

Call DetailsAdvanced Grant (AdG), LS9, ERC-2013-ADG

Summary"Recently through de novo peptide design, we have discovered and presented a new protein structure. This is an all-parallel, 6-helix bundle with a continuous central channel of 0.5 – 0.6 nm diameter. We posit that this is one of a broader class of protein structures that we call the alpha-helical barrels. Here, in three Work Packages, we propose to explore these structures and to develop protein functions within them. First, through a combination of computer-aided design, peptide synthesis and thorough biophysical characterization, we will examine the extents and limits of the alpha-helical-barrel structures. Whilst this is curiosity driven research, it also has practical consequences for the studies that will follow; that is, alpha-helical barrels made from increasing numbers of helices have channels or pores that increase in a predictable way. Second, we will use rational and empirical design approaches to engineer a range of functions within these cavities, including binding capabilities and enzyme-like activities. Finally, and taking the programme into another ambitious area, we will use the alpha-helical barrels to template other folds that are otherwise difficult to design and engineer, notably beta-barrels that insert into membranes to render ion-channel and sensor functions."

"Recently through de novo peptide design, we have discovered and presented a new protein structure. This is an all-parallel, 6-helix bundle with a continuous central channel of 0.5 – 0.6 nm diameter. We posit that this is one of a broader class of protein structures that we call the alpha-helical barrels. Here, in three Work Packages, we propose to explore these structures and to develop protein functions within them. First, through a combination of computer-aided design, peptide synthesis and thorough biophysical characterization, we will examine the extents and limits of the alpha-helical-barrel structures. Whilst this is curiosity driven research, it also has practical consequences for the studies that will follow; that is, alpha-helical barrels made from increasing numbers of helices have channels or pores that increase in a predictable way. Second, we will use rational and empirical design approaches to engineer a range of functions within these cavities, including binding capabilities and enzyme-like activities. Finally, and taking the programme into another ambitious area, we will use the alpha-helical barrels to template other folds that are otherwise difficult to design and engineer, notably beta-barrels that insert into membranes to render ion-channel and sensor functions."